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A. Oblomkov

Publications and source records attributed to A. Oblomkov.

9 recordsLinked to original sources
arXiv2020

Virasoro constraints for stable pairs on toric 3-folds

Using new explicit formulas for the stationary GW/PT descendent correspondence for nonsingular projective toric 3-folds, we show that the correspondence intertwines the Virasoro constraints in Gromov-Witten theory for stable maps with the Virasoro constraints for stable pairs. Since the Virasoro constraints in Gromov-Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric 3-folds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.

math.AG↗
arXiv2019

EGL formula for DT/PT theory of local curves

In this note we prove an integral formula for the bare one-leg PT vertex with descendents. The formula follows from the PT version of Ellingsrud-Göttsche-Lehn formula that is explained here. We apply the integral formula to obtain an elementary proof of rationality of one-leg capped PT vertex with descendents. We also obtain an integral formula for degree zero DT invariants with descendents. Finally we propose an explicit non-equivariant DT/PT correspondence as well as one descendent insertion fully-equivariant DT/PT formula.

math.AG↗
arXiv2009

Quantum cohomology of the Hilbert scheme of points on A_n-resolutions

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra \hat{gl}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A_n x P^1. We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type D and E.

math.AG↗
arXiv2008

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

math.AG↗
arXiv2008

Donaldson-Thomas theory of A_n x P^1

We study the relative Donaldson-Thomas theory of A_n x P^1, where A_n is the surface resolution of a type A_n singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra \hat{gl}(n+1) on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov-Witten theory of A_n x P^1 and the quantum cohomology of the Hilbert scheme of points on A_n.

math.AG↗
arXiv2008

A Lie-theoretic construction of spherical symplectic reflection algebras

We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semi-simple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra.

math.QA↗
arXiv2006

Deformed Harish-Chandra homomorphism for the cyclic quiver

In the case of cyclic quiver we prove that the deformed Harish-Chandra map whose existence was conjectured by Etingof and Ginzburg is well defined. As an application we prove Kirillov-type formula for the cyclotomic Bessel function.

math.RT↗
arXiv2003

Double affine Hecke algebras and Calogero-Moser spaces

In this paper we prove that the spherical subalgebra $eH_{1,τ}e$ of the double affine Hecke algebra $H_{1,τ}$ is an integral Cohen-Macaulay algebra isomorphic to the center $Z$ of $H_{1,τ}$, and $H_{1,τ}e$ is a Cohen-Macaulay $eH_{1,τ}e$-module with the property $H_{1,τ}=End_{eH_{1,τ}e}(H_{1,τ}e)$. In the case of the root system $A_{n-1}$ the variety $Spec(Z)$ is smooth and coincides with the completion of the configuration space of the relativistic analog of the trigomonetric Calogero-Moser system. This implies the result of Cherednik that the module $eH_{1,τ}$ is projective and all irreducible finite dimensional representations of $H_{1,τ}$ are regular representation of the finite Hecke algebra.

math.RT↗